Sunday, January 07, 2018 - Saturday, January 13, 2018

Colloquium

Title: Elliptic curves and p-adic L-functions

Date: 01/08/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: David Hansen, Columbia University

I'll explain the notion of a p-adic L-function, try to
motivate why one might care about such a gadget, and give some history of their construction and applications. At the end of the talk I'll discuss a recent joint work with John Bergdall in which (among other things) we construct canonical p-adic L-functions associated with modular elliptic curves over totally real number fields.

Observation data along with mathematical models play a crucial role in improving prediction skills in science and engineering. In this talk we focus on the recent development of uncertainty quantification methods, data assimilation and parameter estimation, for Physics-constrained problems that are often described by partial differential equations. We discuss the similarities shared by the two methods and their differences in mathematical and computational points of view and future research topics. As applications, numerical weather prediction for geophysical flows and parameter estimation of kinetic reaction rates in the hydrogen-oxygen combustion are provided.

Colloquium

Title: Quantifying congruences between Eisenstein series and cusp forms

Date: 01/10/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Preston Wake, UCLA

Consider the following two problems in algebraic number theory:
1. For which prime numbers p can we easily show that the Fermat equation x^p + y^p =z^p has no non-trivial integer solutions?
2. Given an elliptic curve E over the rational numbers, what can be said about the group of rational points of finite order on E?
These seem like very different problems, but, surprisingly, they share a common theme: they are both related to the existence of congruences between two types of modular forms, Eisenstein series and cusp forms. We will explain these examples, and discuss a new technique for giving quantitative information about these congruences (for example, counting the number of cusp forms congruent to an Eisenstein series). We will explain how this can give finer arithmetic information than simply knowing the existence of a congruence. This is joint work with Carl Wang-Erickson.